This result, as opposed to classical variational regularization theory e.g. [1]}, has the advantage that at no point do we require exact global minimizers of the functionals \(\mathcal {T}_{\alpha , y}\)  but only approximate minimizers, which may be more easily reachable in practical applications.
Consider for example the case where we employ an iterative algorithm which has convergence guarantees of the form \(\mathcal {F}(x_n) - \mathcal {F}(x_*) \le C / n^r\)  for the \(n\) -th iterate and \(x_*\)  being a minimizer of \(\mathcal {F}\) . Applying this algorithm to \(\mathcal {F}= \mathcal {T}_{\alpha , y}\)  and requiring that \(C / n^r \le \alpha \varepsilon \)  in order to get that \(x_n\)  is an \(\alpha \phi \) -critical point, we see that the above theory shows that one might stop the iterative algorithm after a finite amount of steps, i.e. we do not necessarily need to run the algorithm until it converges and we still get a stable and convergent regularization method.
